Upper and lower bound results on the convex hull of integer points in polyhedra

Abstract

In typical linear programming problems, we are concerned with finding non-negative integers {x1,…, xn} that maximize a linear form c1x1 + … + cnxn, subject to a number of linear inequalities, for The maximum is necessarily attained at one of the vertices of the convex hull of integer points defined by the inequalities, so we have an interest in estimating the number M of these vertices. We give two results; one improving an upper bound result for M of Hayes and Larman concerning the Knapsack polytope, the other an example showing that, in 3-dimensions, it is possible to choose the coefficients aij to obtain a lower bound for M.